We explore the statistical aspects of some of the known methods of analysing experts’ elicited data to identify potential improvements on the accuracy of their outcomes in this study. It can be identified that potential correlation structures induced in the probability predictions by the characteristics of experimental designs are ignored in computing experts’ Brier scores. We show that the accuracy of the standard error estimates of experts’ Brier scores can be improved by incorporating the within-question correlations of probability predictions in the second chapter of this thesis. Missing probability predictions of events can impact on assessing the prediction accuracy of experts using different sets of events (Merkle et al., 2016; Hanea et al., 2018). It is shown in the third chapter that multiple imputation method using a mixed-effects model with questions’ effects as random effects can effectively estimate missing predictions to enhance the comparability of experts’ Brier scores. Testing experts’ calibration on eliciting credible intervals of unknown quantities using hit rates; observed proportions of elicited intervals that contain realized values of given quantities (McBride, Fidler, and Burgman, 2012), has a property of obtaining lower values of power to correctly identify well-calibrated experts and more importantly, the power tends to decrease as the number of elicited intervals increases. The equivalence test of a single binomial proportion can be used to overcome these problems as shown in the fourth chapter. There is a possibility of allocating higher weights to some of the not well-calibrated experts by the way experts’ calibration is assessed in the Cooke’s classical model (Cooke, 1991) to derive experts’ weights. We show that the multinomial equivalence test can be used to overcome this problem in the fifth chapter. Experts’ weights that derived from experiments to combine experts’ elicited subjective probability distributions to obtain aggregated probability distributions of unknown quantities (O’Hagan, 2019) are random variables subject to uncertainty. We derive shrinkage experts’ weights with reduced mean squared errors in the sixth chapter to enhance the precision of the resulting aggregated distributions of quantities.

Streaming data are a type of high-frequency and nonstationary time series data. The collection of streaming data is sequential and potentially never-ending. Examples of streaming data, including data from sensor networks, mobile devices and the Internet, are prevalent in our daily lives. An estimator for streaming data needs to be computationally efficient so that it is relatively easy to update the estimator using newly arrived data. In addition, the estimator has to be adaptive to the nonstationarity of data. These constraints make streaming data analysis more challenging than analysing the conventional non-streaming data sets. Although streaming data analysis has been discussed in the machine learning community for more than two decades, it has received limited attention from statistical researchers. Estimation methods that are both computationally efficient and theoretically justified are still lacking. In this thesis, we propose nonparametric density and regression estimation methods for streaming data, where the smoothing parameters are chosen in a computationally efficient and fully data-driven way. These methods extend some classical kernel smoothing techniques, such as the kernel density estimator and the Nadaraya-Watson regression estimator, to address the theoretical and computational challenges arising from streaming data analysis. Asymptotic analyses provide these methods with theoretical justification. Numerical studies have shown the superiority of our methods over conventional ones. Through some real-data examples, we show that these methods are potentially useful in modelling real-world problems. Finally, we discuss some directions for future research, including extending these methods to model higher-dimensional streaming data and to streaming data classification.

Optimisation algorithms require careful tuning and analysis to perform well in practice. Their performance is strongly affected by algorithm parameter choices, software, and hardware and must be analysed empirically. To conduct such analysis, researchers and developers require high-quality libraries of test instances. Improving the diversity of these test sets is essential to driving the development of well-tested algorithms. This thesis is focused on producing synthetic test sets for Mixed Integer Programming (MIP) solvers. Synthetic data should be carefully designed to be unbiased, diverse with respect to measurable features of instances, have tunable properties to replicate real-world problems, and challenge the vast array of algorithms available. This thesis outlines a framework, methods and algorithms developed to ensure these requirements can be met with synthetically generated data for a given problem. Over many years of development, MIP solvers have become increasingly complex. Their overall performance depends on the interactions of many different components. To cope with this complexity, we propose several extensions over existing approaches to generating optimisation test cases. First, we develop alternative encodings for problem instances which restrict consideration to relevant instances. This approach provides more control over instance features and reduces the computational effort required when we have to resort to search-based generation approaches. Second, we consider more detailed performance metrics for MIP solvers in order to produce test cases which are not only challenging but from which useful insights can be gained. This work makes several key contributions: 1. Performance metrics are identified which are relevant to component algorithms in MIP solvers. This helps to define a more comprehensive performance metric space which looks beyond benchmarking statistics such as CPU time required to solve a problem. Using these more detailed performance metrics we aim to produce explainable and insightful predictions of algorithm performance in terms of instance features. 2. A framework is developed for encoding problem instances to support the design of new instance generators. The concepts of completeness and correctness defined in this framework guide the design process and ensure all problem instances of potential interest are captured in the scheme. Instance encodings can be generalised to develop search algorithms in problem space with the same guarantees as the generator. 3. Using this framework new generators are defined for LP and MIP instances which control feasibility and boundedness of the LP relaxation, and integer feasibility of the resulting MIP. Key features of the LP relaxation solution, which are directly controlled by the generator, are shown to affect problem difficulty in our analysis of the results. The encodings used to control these properties are extended into problem space search operators to generate further instances which discriminate between solver configurations. This work represents the early stages of an iterative methodology required to generate diverse test sets which continue to challenge the state of the art. The framework, algorithms and codes developed in this thesis are intended to support continuing development in this area.

Rapidly growing demand and soaring costs for healthcare services in Australia and across the world are jeopardising the sustainability of government-funded healthcare systems. We need to be innovative and more efficient in delivering healthcare services in order to keep the system sustainable. In this thesis, we utilise a number of scientific tools to improve the patient flow in a surgical suite of a hospital and subsequently develop a structured approach for intelligent patient flow management. First, we analyse and understand the patient flow process in a surgical suite. Then we obtain data from the partner hospital and extract valuable information from a large database. Next, we use machine learning techniques, such as classification and regression tree analysis, random forest, and k-nearest neighbour regression, to classify patients into lower variability resource user groups and fit discrete phase-type distributions to the clustered length of stay data. We use length of stay scenarios sampled from the fitted distributions in our sequential stochastic mixed-integer programming model for tactical master surgery scheduling. Our mixed-integer programming model has the particularity that the scenarios are utilised in a chronologically sequential manner, not in parallel. Moreover, we exploit the randomness in the sample path to reduce the requirement of optimising the process for many scenarios which helps us obtain high-quality schedules while keeping the problem algorithmically tractable. Last, we model the patient flow process in a healthcare facility as a stochastic process and develop a model to predict the probability of the healthcare facility exceeding capacity the next day as a function of the number of inpatients and the next day scheduled patients, their resource user groups, and their elapsed length of stay. We evaluate the model's performance using the receiver operating characteristic curve and illustrate the computation of the optimal threshold probability by using cost-benefit analysis that helps the hospital management make decisions.

Modelling of spatio-temporal count data has received considerable attention in recent statistical research. However, the presence of massive correlation between locations, time points and variables imposes a great computational challenge. In existing literature, latent models under the Bayesian framework are predominately used. Despite numerous theoretical and practical advantages, likelihood analysis of spatio-temporal modelling on count data is less wide spread, due to the difficulty in identifying the general class of multivariate distributions for discrete responses. In this thesis, we propose a Gaussian copula regression model (copSTM) for the analysis of multivariate spatio-temporal data on lattice. Temporal effects are modelled through the conditional marginal expectations of the response variables using an observation-driven time series model, while spatial and cross-variable correlations are captured in a block dependence structure, allowing for both positive and negative correlations. The proposed copSTM model is flexible and sufficiently generalizable to many situations. We provide pairwise composite likelihood inference tools. Numerical examples suggest that the proposed composite likelihood estimator produces satisfactory estimation performance. While variable selection of generalized linear models is a well developed topic, model subsetting in applications of Gaussian copula models remains a relatively open research area. The main reason is the computational burden that is already quite heavy for simply fitting the model. It is therefore not computationally affordable to evaluate many candidate sub-models. This makes penalized likelihood approaches extremely inefficient because they need to search through different levels of penalty strength, apart from the fact suggested by our numerical experience that optimization of penalized composite likelihoods with many popular penalty terms (e.g LASSO and SCAD) usually does not converge in copula models. Thus, we propose to use a criterion-based selection approach that borrows strength from the Gibbs sampling technique.The methodology guarantees to converge to the model with the lowest criterion value, yet without searching through all possible models exhaustively. Finally, we present an R package implementing the estimation and selection of the copSTM model in C++. We show examples comparing our package to many available R packages (on some special cases of the copSTM), confirming the correctness and efficiency of the package functions. The package copSTM provides a competitive toolkit option for the analysis spatio-temporal count data on lattice in terms of both model flexibility and computational efficiency.

This thesis presents the computation of singular vectors of the W_n algebras and the BRST cohomology of modules of the simple vertex operator algebra L_k(sl2) associated to the affine Lie algebra of sl2 in the relaxed category We will first recall some general theory on vertex operator algebras. We will then introduce the module categories that are relevant for conformal field theory. They are the category O of highest-weight modules and the relaxed category which contains O as well as the relaxed highest-weight modules with spectral flow and non-split extensions. We will then introduce the W_n algebras and the simple vertex operator algebra L_k(sl2). Properties of the Heisenberg algebra, the bosonic and the fermionic ghosts will be discussed as they are required in the free field realisations of W_n and L_k(sl2) as well as the construction of the BRST complex. We will then compute explicitly the singular vectors of W_n algebras in their Fock representations. In particular, singular vectors can be realised as the image of screening operators of the W_n algebras. One can then realise screening operators in terms of Jack functions when acting on a highest-weight state, thereby obtaining explicit formulae of the singular vectors in terms of symmetric functions. We will then discuss the BRST construction and the BRST cohomology for modules in category O. Lastly we compute the BRST cohomology for L_k(sl2) modules in the relaxed category. In particular, we compute the BRST cohomology for the highest-weight modules with positive spectral flow for all degrees and the BRST cohomology for the highest-weight modules with negative spectral flow for one degree.

This thesis examines three problems in statistics: the missing data problem in the context of extracting trends from time series data, the combinatorial model selection problem in regression analysis, and the structure learning problem in graphical modelling / system identification. The goal of the first problem is to study how uncertainty in the missing data affects trend extraction. This work derives an analytical bound to characterise the error of the estimated trend in terms of the error of the imputation. It works for any imputation method and various trend-extraction methods, including a large subclass of linear filters and the Seasonal-Trend decomposition based on Loess (STL). The second problem is to tackle the combinatorial complexity which arises from the best-subset selection in regression analysis. Given p variables, a model can be formed by taking a subset of the variables, and the total number of models p is $2^p$. This work shows that if a hierarchical structure can be established on the model space, then the proposed algorithm, Gibbs Stochastic Search (GSS), can recover the true model with probability one in the limit and high probability with finite samples. The core idea is that when a hierarchical structure exists, every evaluation of a wrong model would give information about the correct model. By aggregating these information, one may recover the correct model without exhausting the model space. As an extension, parallelisation of the algorithm is also considered. The third problem is about inferring from data the systemic relationship between a set of variables. This work proposes a flexible class of multivariate distributions in a form of a directed acyclic graphical model, which uses a graph and models each node conditioning on the rest using a Generalised Linear Model (GLM), and it shows that while the number of possible graphs is $\Omega(2^{p \choose 2})$, a hierarchical structure exists and the GSS algorithm applies. Hence, a systemic relationship may be recovered from the data. Other applications like imputing missing data and simulating data with complex covariance structure are also investigated.

The exclusion process has been the default model for the transportation phenomenon. One fundamental issue is to compute the exact formulae analytically. Such formulae enable us to obtain the limiting distribution through asymptotics analysis, and they also allow us to uncover relationships between different processes, and even between very different systems. Extensive results have been reported for single-species systems, but few for multi-component systems and mixtures. In this thesis, we focus on multi-species exclusion processes, and propose two approaches for exact solutions. The first one is due to duality, which is defined by a function that co-varies in time with respect to the evolution of two processes. It relates physical quantities, such as the particle flow, in a system with many particles to one with few particles, so that the quantity of interest in the first process can be calculated explicitly via the second one. Historically, published dualities have mostly been found by trial and error. Only very recently have attempts been made to derive these functions algebraically. We propose a new method to derive dualities systematically, by exploiting the mathematical structure provided by the deformed quantum Knizhnik-Zamolodchikov equation. With this method, we not only recover the well-known self-duality in single-species asymmetric simple exclusion processes (ASEPs), and also obtain the duality for two-species ASEPs. Solving the master equation is an alternative method. We consider an exclusion process with 2 species particles: the AHR (Arndt-Heinzl-Rittenberg) model and give a full derivation of its Green's function via coordinate Bethe ansatz. Hence using the Green's function, we obtain an integral formula for its joint current distributions, and then study its limiting distribution with step type initial conditions. We show that the long-time behaviour is governed by a product of the Gaussian and the Gaussian unitary ensemble (GUE) Tracy-Widom distributions, which is related to the random matrix theory. Such result agrees with the prediction made by the nonlinear fluctuating hydrodynamic theory (NLFHD). This is the first analytic verification of the prediction of NLFHD in a multi-species system.

The main objects of investigation in this thesis are two Yang-Baxter integrable lattice models of statistical mechanics in two dimensions: nonunitary RSOS models and dimers. At criticality they admit continuum descriptions with nonunitary conformal field theories (CFTs) in (1+1) dimensions. CFTs are quantum field theory invariant under conformal transformations. They play a major role in the theory of phase transition and critical phenomena. In quantum field theory unitarity is the requirement that the probability is conserved, hence realistic physical problems are associated with unitary quantum field theories. Nevertheless, in statistical mechanics this property loses a physical meaning and statistical systems like polymers and percolations, which model physical problems with long-range interactions, in the continuum scaling limit give rise to nonunitary conformal field theories. Both the nonunitary RSOS models and dimers are defined on a two-dimensional square lattice. Restricted solid-on-solid (RSOS) models are so called because their degrees of freedom are in the form of a finite (therefore restricted) set of heights which live on the sites of the lattice and their interactions take place between the four sites around each face of the lattice (solid-on-solid). Each allowed configuration of heights maps to a specific Boltzmann weight. RSOS are integrable in the sense that their Boltzmann weights and transfer matrices satisfy the Yang-Baxter equation. The CFTs associated to critical RSOS models are minimal models, the simplest family of rational conformal field theories. The process of fusion on elementary RSOS models has a different outcome on the CFT side depending on both the level of fusion and the value of their crossing parameter λ. Precisely, in the interval 0 < λ < π/2, the 2x2 fused RSOS models correspond to higher-level conformal cosets with integer level of fusion equal to two. Instead in the complementary interval π/n < λ < π the 2x2 fused RSOS models are related to minimal models with integer level of fusion equal to one. To prove this conjecture one-dimensional sums, deriving from the well-known Yang-Baxter corner transfer matrix method, have been calculated, extended in the continuum limit and ultimately compared to the conformal characters of the related minimal models. The dimer model has been for a long time object of various scientific studies, firstly as simple prototype of diatomic molecules and then as equivalent formulation to the well-known domino tilings problem of statistical mechanics. However, only more recently it has attracted attention as conformal field theory thanks to its relation with another famous integrable lattice model, the six-vertex model. What is particularly interesting of dimers is the property of being a free-fermion model and at the same time showing non-local properties due to the long-range steric effects propagating from the boundaries. This non-locality translates then in the dependance of their bulk free energy on the boundary conditions. We formulate the dimer model as a Yang-Baxter integrable free-fermion six-vertex model. This model is integrable in different geometries (cylinder, torus and strip) and with a variety of different integrable boundary conditions. The exact solution for the partition function arises from the complete spectra of eigenvalues of the transfer matrix. This is obtained by solving some functional equations, in the form of inversion identities, usually associated to the transfer matrix of the free-fermion six-vertex model, and using the physical combinatorics of the pattern of zeros of the transfer matrix eigenvalues to classify the eigenvalues according to their degeneracies. In the case of the cylinder and torus, the transfer matrix can be diagonalized, while, in the other cases, we observe that in a certain representation the double row transfer matrix exhibits non trivial Jordan-cells. Remarkably, the spectrum of eigenvalues of dimers and critical dense polymers agree sectors by sectors. The similarity with critical dense polymers, which is a logarithmic field theory, raises the question whether also the free-fermion dimer model manifests a logarithmic behaviour in the continuum scaling limit. The debate is still open. However, in our papers we provide a final answer and argue that the type of conformal field theory which best describe dimers is a logarithmic field theory, as it results by looking at the numerically estimate of the finite size corrections to the critical free energy of the free-fermion six-vertex-equivalent dimer model. The thesis is organized as follows. The first chapter is an introduction which has the purpose to inform the reader about the basics of statistical mechanics, from one side, and CFTs, on the other side, with a specific focus on the two lattice models that have been studied (nonunitary RSOS and dimers) and the theories associated to the their continuum description at criticality (minimal models and logarithmic CFTs). The second chapter considers the family of non-unitary RSOS models with π/n < λ < π and brings forward the discussion around the one-dimensional sums of the elementary and fused models, and the associated conformal characters in the continuum scaling limit. The third and fourth chapters are dedicated to dimers, starting with periodic conditions on a cylinder and torus, and then more general integrable boundary conditions on a strip. In each case, a combinatorial analysis of the pattern of zeros of the transfer matrix eigenvalues is presented and extensively treated. It follows then the analysis of the finite-size corrections to the critical free energy. Finally, the central charges and minimal conformal dimensions of critical dimers are discussed in depth with concluding remarks about the logarithmic hypothesis. Next, there is a conclusion where the main results of these studies are summarized and put into perspective with possible future research goals.

Freight distribution systems are under stress. With the world population growing, the migration of people to urban areas and technologies that allow purchases from virtually anywhere, efficient freight distribution can be challenging. An inefficient movement of goods may lead to business not being economically viable and also has social and environmental negative effects. An important strategy to be incorporated in freight distribution systems is the consolidation of goods, i.e., group goods by their destination. This strategy increases vehicles utilisation, reducing the number of vehicles and the number of trips required for the distribution and, consequently, costs, traffic, noise and air pollution. In this thesis, we explore consolidation in three different contexts (or cases) from an optimisation point of view. Each context is related to optimisation problems for which we developed mathematical programming models and solution methods. The first case in which we explore consolidation is in container loading problems (CLPs). CLPs are a class of packing problems which aims at positioning three-dimensional boxes inside a container efficiently. The literature has incorporated many practical aspects into container loading solution method (e.g. restricting orientation of boxes, stability and weight distribution). However, to the best of our knowledge, the case considering more dynamic systems (e.g. cross-docking) in which goods might have a schedule of arrival were yet to be contemplated by the literature. We define an extension of CLP which we call Container Loading Problem with Time Availability Constraints (CLPTAC), which considers boxes are not always available for loading. We propose an extension of a CLP model that is suitable for CLPTAC and solution methods which can also handle cases with uncertainty in the schedule of the arrival of the boxes. The second case is a more broad view of the network, considering an open vehicle routing problem with cross-dock selection. The traditional vehicle routing problem has been fairly studied. Its open version (i.e. with routes that start and end at different points) has not received the same attention. We propose a version of the open vehicle routing problem in which some nodes of the network are consolidation centres. Instead of shippers sending goods directly to their consumers, they must send to one of the available consolidation centres, then, goods are resorted and forwarded to their destination. For this problem, we propose a mixed integer linear programming model for cost minimisation and a solution method based on the Benders decomposition framework. A third case in which we explored consolidation is in collaborative logistics. Particularly, we focus on the shared use of the currently available infrastructure. We defined a hubselection problem in which one of the suppliers is selected as a hub. In a hub facility, other suppliers might meet to exchange their goods allowing one supplier to satisfy the demand from others. For this problem, we propose a mixed integer linear programming model and a heuristic based on the model. Moreover, we compared a traditional distribution strategy, with each supplier handling its demand, against the collaborative one. In this thesis, we explore these three cases which are related to consolidation for improving the efficiency in freight distribution systems. We extend some problems (e.g. versions of CLP) to apply them to a more dynamic setting and we also define optimisation problems for networks with consolidation centres. Furthermore, we propose solution methods for each of the defined problems and evaluate them using randomly generated instances, benchmarks from the literature and some cases based on real-world characteristics.

Despite advances in prevention and control, infectious diseases continue to be a burden to human health. Many factors, including the waning and boosting of immunity, are involved in the spread of disease. The link between immunological processes at an individual level and population-level immunity is complex and subtle. Mathematical models are a useful tool to understand the biological mechanisms behind the observed epidemiological patterns of an infectious disease. Here I construct deterministic, compartment models of infectious disease transmission to study the effects of waning and boosting of immunity on infectious disease dynamics. While waning immunity has been studied in many models, incorporation of immune boosting in models of transmission has largely been neglected. In my study, I look at three different aspects of immune boosting: (i) the influence of immune boosting on the critical vaccination thresholds required for disease control; (ii) the effect of immune boosting and cross-immunity on infectious disease dynamics; and (iii) the influence of differentiating vaccine-acquired immunity from natural (infection-acquired) immunity and different mechanisms of immune boosting on infection prevalence. Models can provide support for public health control measures in terms of critical vaccination thresholds. There is a direct relationship, from mathematical theory, between the critical vaccination threshold and the basic reproduction number, R0. Key epidemiological quantities, such as R0, are measured from data, but the selection of the model used to infer these quantities matters. I show how the inferred values of R0---and thus, critical vaccination thresholds---can vary when immune boosting is taken into account. I also investigate the effects of interactions between immune boosting and cross-immunity on infectious disease dynamics, using a two-pathogen transmission model. Immunity to one pathogen that confers immunity to another pathogen, or to another strain of a given pathogen, is known as cross-immunity. Varying levels of susceptibility to infection conferred by cross-immunity are included in the model. Using a combination of numerical simulations and bifurcation analyses, I show that immune boosting at strong levels can lead to recurrent epidemics (or periodic solutions) independent of cross-immunity. Where immune boosting is weak, cross-immunity allows the model to generate periodic solutions. For some diseases, there are differences in infection-acquired immunity and vaccine-acquired immunity. I explore the effect of vaccination and immune boosting on epidemiological patterns of infectious disease. I construct and analyse a model that differentiates vaccine-acquired immunity from infection-acquired immunity in the form of duration of protection. The model also distinguishes between primary and secondary infections. I show that vaccination is effective at reducing primary infections but not necessarily secondary infections, which can maintain overall transmission. Two different mechanisms through which immune boosting provides protection are also explored. Whether immune boosting delays or bypasses a primary infection can determine whether primary or secondary infections contribute most to transmission.

Influenza inflicts a substantial burden on society but accurate and timely forecasts of seasonal epidemics can help mitigate this burden by informing interventions to reduce transmission. Recently, both statistical (correlative) and mechanistic (causal) models have been used to forecast epidemics. However, since mechanistic models are based on the causal process underlying the epidemic they are poised to be more useful in the design of intervention strategies. This study investigate approaches to improve epidemic forecasting using mechanistic models. In particular, it reports on efforts to improve a forecasting system targeting seasonal influenza epidemics in major cities across Australia. To improve the forecasting system we first needed a way to benchmark its performance. We investigate model selection in the context of forecasting, deriving a novel method which extends the notion of Bayes factors to a predictive setting. Applying this methodology we found that accounting for seasonal variation in absolute humidity improves forecasts of seasonal influenza in Melbourne, Australia. This result holds even when accounting for the uncertainty in predicting seasonal variation in absolute humidity. Our initial attempts to forecast influenza transmission with mechanistic models were hampered by high levels of uncertainty in forecasts produced early in the season. While substantial uncertainty seems inextricable from long-term prediction, it seemed plausible that historical data could assist in reducing this uncertainty. We define a class of prior distributions which simplify the process of incorporating existing knowledge into an analysis, and in doing so offer a refined interpretation of the prior distribution. As an example we used historical time series of influenza epidemics to reduce initial uncertainty in forecasts for Sydney, Australia. We explore potential pitfalls that may be encountered when using this class of prior distribution. Deviating from the theme of forecasting, we consider the use of branching processes to model early transmission in an epidemic. An inhomogeneous branching process is derived which allows the study of transmission dynamics early in an epidemic. A generation dependent offspring distribution allows for the branching process to have sub-exponential growth on average. The multi-scale nature of a branching process allows us to utilise both time series of incidence and infection networks. This methodology is applied to data collected during the 2014–2016 Ebola epidemic in West-Africa leading to the inference that transmission grew sub-exponentially in Guinea, Liberia and Sierra Leone. Throughout this thesis, we demonstrate the utility of mechanistic models in epidemiology and how a Bayesian approach to statistical inference is complementary to this.